Re: [Phys-L] Half-Life measurement : uncertainties, correlations, SVD

[John Denker <jsd@av8n.com>]

On 10/19/21 10:35 AM, Paul Nord wrote:

> My problem with that is that the data are correlated to each other.

[snip]

> Would it be better to consider only the differential counts between
> time t1 and t2?

Well, that's what my fitting function does.

Here's how I think about it: The term "Poisson" refers to a one-
parameter family of distributions. The parameter is often denoted
λ. Sometimes called the intensity. The shape of the distribution
tells you the probability of getting k events during an interval
with intensity λ. It turns out that the mean of the distribution
is equal to λ, so sometimes λ is just called the mean.

I mention this because if you know k and λ, the distribution does
not care about the timing at all. You could have λ=100 k=99 spread
over 1 second or spread over 1 fortnight; the distribution doesn't
care.

Each observation in the data file comes from a cumulative timer and
a cumulative counter. My code subtracts two consecutive counter
values to get k, i.e. the observed number of events in the interval.
It integrates the decay laws to find the model prediction for λ,
i.e. the mean number of counts there "should" be in the interval.

This requires doing the integral, which is why I posted some gory
details about integrating exponentials. Let's be clear: I never
evaluate the Poisson probability "at" the time of any observation.
All the time dependence is inside the integrand. The integral tells
me the intensity λ for the interval.

To repeat: My model analyzes the situation interval by interval.
In some sense it knows the global shape of the decay curve, but
it doesn't need to know that; it doesn't directly care about that.
In particular: The counter does not need to start at zero. The
analysis is strictly gauge invariant with respect to the initial
counter value.

The same is "almost" true of the initial clock value. The amplitudes
are quoted at time t=0. If you shift the time you have to rescale
the amplitudes accordingly. So there is time symmetry also, just a
slightly complicated symmetry.

> the sum of Poisson distributions is still a Poisson distribution.

That's true and important. Note that we are talking about a weighted
sum, which has to be done properly. That's what the integral does.

===========

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